The point of introducing Krawtchouk polynomials is that we can bring in asymptotic expansions of Krawtchouk polynomials. From [1]} (see also [2]}, [3]}) we obtain

Let \(n,t\)  and \(w\)  be three positive integers.
We set \(\tau \mathop {=}\limits ^{\triangle }\frac{t}{n}\) , \(\omega \mathop {=}\limits ^{\triangle }\frac{w}{n}\)  and \(\omega ^\perp \mathop {=}\limits ^{\triangle }\frac{1}{2}-\sqrt{\omega (1-\omega )}\) .
We assume \(t \le n/2\) . Let \(z \mathop {=}\limits ^{\triangle }\frac{1-2\tau - \sqrt{D}}{2 (1-\omega )}\)  where
\(D \mathop {=}\limits ^{\triangle }\left(1- 2 \tau \right)^2-4 \omega (1-\omega )\) .
In the case \(\tau \in [0,\omega ^\perp ]\) ,
\(K_{w}^n(t) = \sqrt{\frac{1-z^2}{2 \pi \sqrt{D}n}} \;\;\frac{(1-z)^t(1+z)^{n-t}}{z^{w+\frac{1}{2}}}\;( 1 + O(w^{-1})).\) 
